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Chapter 9: Q1 (page 437)

Independent and Dependent Samples Which of the following involve independent samples?

a. Data Set 14 “Oscar Winner Age” in Appendix B includes pairs of ages of actresses and actors at the times that they won Oscars for Best Actress and Best Actor categories. The pair of ages of the winners is listed for each year, and each pair consists of ages matched according to the year that the Oscars were won.

b. Data Set 15 “Presidents” in Appendix B includes heights of elected presidents along with the heights of their main opponents. The pair of heights is listed for each election.

c. Data Set 26 “Cola Weights and Volumes” in Appendix B includes the volumes of the contents in 36 cans of regular Coke and the volumes of the contents in 36 cans of regular Pepsi.

Short Answer

Expert verified

a. It is a dependent sample.

b. It is a dependent sample.

c. It is an independent sample.

Step by step solution

01

Define independent and dependent samples

Independent samples were taken from two populations. The values are not associated with one other.

Dependent samples have a relationship between values such that each value of one sample is related to another value of another sample.

02

Identify the samples

a. Any pair of actors and actress is related by the year in which they won the Oscars. Hence, each pair consists of ages matched according to the year or time. Thus, the samples are dependent.

b. Any pair of presidents and opponents recorded for height are related by the year of election. Thus, the samples are dependent.

c. The two samples for contents of 36 cans in each group, Pepsi and Cokeare not related to each other. Thus, the samplesare independent.

Only part (c) describes independent samples.

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Most popular questions from this chapter

A sample size that will ensure a margin of error of at most the one specified.

Determining Sample Size The sample size needed to estimate the difference between two population proportions to within a margin of error E with a confidence level of 1 - a can be found by using the following expression:

\({\bf{E = }}{{\bf{z}}_{\frac{{\bf{\alpha }}}{{\bf{2}}}}}\sqrt {\frac{{{{\bf{p}}_{\bf{1}}}{{\bf{q}}_{\bf{1}}}}}{{{{\bf{n}}_{\bf{1}}}}}{\bf{ + }}\frac{{{{\bf{p}}_{\bf{2}}}{{\bf{q}}_{\bf{2}}}}}{{{{\bf{n}}_{\bf{2}}}}}} \)

Replace \({{\bf{n}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{n}}_{\bf{2}}}\) by n in the preceding formula (assuming that both samples have the same size) and replace each of \({{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{q}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}}\;{\bf{and}}\;{{\bf{q}}_{\bf{2}}}\)by 0.5 (because their values are not known). Solving for n results in this expression:

\({\bf{n = }}\frac{{{\bf{z}}_{\frac{{\bf{\alpha }}}{{\bf{2}}}}^{\bf{2}}}}{{{\bf{2}}{{\bf{E}}^{\bf{2}}}}}\)

Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want 95% confidence that your error is no more than 0.03.

Using Confidence Intervals

a. Assume that we want to use a 0.05 significance level to test the claim that p1 < p2. Which is better: A hypothesis test or a confidence interval?

b. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method; P-value method; critical value method?

c. If we want to use a 0.05 significance level to test the claim that p1 < p2, what confidence level should we use?

d. If we test the claim in part (c) using the sample data in Exercise 1, we get this confidence interval: -0.000508 < p1 - p2 < - 0.000309. What does this confidence interval suggest about the claim?

Family Heights. In Exercises 1–5, use the following heights (in.) The data are matched so that each column consists of heights from the same family.

1. a. Are the three samples independent or dependent? Why?

b. Find the mean, median, range, standard deviation, and variance of the heights of the sons.

c. What is the level of measurement of the sample data (nominal, ordinal, interval, ratio)?

d. Are the original unrounded heights discrete data or continuous data?

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of\({n_1} - 1\)and\({n_2} - 1\).)

Are Male Professors and Female Professors Rated Differently?

a. Use a 0.05 significance level to test the claim that two samples of course evaluation scores are from populations with the same mean. Use these summary statistics: Female professors:

n = 40, \(\bar x\)= 3.79, s = 0.51; male professors: n = 53, \(\bar x\) = 4.01, s = 0.53. (Using the raw data in Data Set 17 “Course Evaluations” will yield different results.)

b. Using the summary statistics given in part (a), construct a 95% confidence interval estimate of the difference between the mean course evaluations score for female professors and male professors.

c. Example 1 used similar sample data with samples of size 12 and 15, and Example 1 led to the conclusion that there is not sufficient evidence to warrant rejection of the null hypothesis.

Do the larger samples in this exercise affect the results much?
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